R-3 Repository :: Browsing by Author "Cox, Dennis D."

Browsing by Author "Cox, Dennis D."

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A statistical model for removing inter-device differences in spectroscopy
(Optical Society of America, 2014) Wang, Lu; Lee, Jong Soo; Lane, Pierre; Atkinson, E. Neely; Zuluaga, Andres; Follen, Michele; MacAulay, Calum; Cox, Dennis D.
We are investigating spectroscopic devices designed to make in vivo cervical tissue measurements to detect pre-cancerous and cancerous lesions. All devices have the same design and ideally should record identical measurements. However, we observed consistent differences among them. An experiment was designed to study the sources of variation in the measurements recorded. Here we present a log additive statistical model that incorporates the sources of variability we identified. Based on this model, we estimated correction factors from the experimental data needed to eliminate the inter-device variability and other sources of variation. These correction factors are intended to improve the accuracy and repeatability of such devices when making future measurements on patient tissue.
Accuracy of optical spectroscopy for the detection of cervical intraepithelial neoplasia without colposcopic tissue information; a step toward automation for low resource settings
(Society of Photo-Optical Instrumentation Engineers, 2012-04) Yamal, Jose-Miguel; Zewdie, Getie A.; Cox, Dennis D.; Atkinson, E. Neely; Cantor, Scott B.; MacAulay, Calum; Davies, Kalatu; Adewole, Isaac; Buys, Timon P. H.; Follen, Michele
Optical spectroscopy has been proposed as an accurate and low-cost alternative for detection of cervical intraepithelial neoplasia. We previously published an algorithm using optical spectroscopy as an adjunct to colposcopy and found good accuracy (sensitivity ¼ 1.00 [95% confidence interval ðCIÞ ¼ 0.92 to 1.00], specificity ¼ 0.71 [95% CI ¼ 0.62 to 0.79]). Those results used measurements taken by expert colposcopists as well as the colposcopy diagnosis. In this study, we trained and tested an algorithm for the detection of cervical intraepithelial neoplasia (i.e., identifying those patients who had histology reading CIN 2 or worse) that did not include the colposcopic diagnosis. Furthermore, we explored the interaction between spectroscopy and colposcopy, examining the importance of probe placement expertise. The colposcopic diagnosis-independent spectroscopy algorithm had a sensitivity of 0.98 (95% CI ¼ 0.89 to 1.00) and a specificity of 0.62 (95% CI ¼ 0.52 to 0.71). The difference in the partial area under the ROC curves between spectroscopy with and without the colposcopic diagnosis was statistically significant at the patient level (p ¼ 0.05) but not the site level (p ¼ 0.13). The results suggest that the device has high accuracy over a wide range of provider accuracy and hence could plausibly be implemented by providers with limited training.
An Alternative Approach for Estimating the Accuracy of Colposcopy in Detecting Cervical Precancer
(Public Library of Science, 2015) Davies, Kalatu R.; Cantor, Scott B.; Cox, Dennis D.; Follen, Michele
Introduction: Since colposcopy helps to detect cervical cancer in its precancerous stages, as new strategies and technologies are developed for the clinical management of cervical neoplasia, precisely determining the accuracy of colposcopy is important for characterizing its continued role. Our objective was to employ a more precise methodology to estimate of the accuracy of colposcopy to better reflect clinical practice. Study design: For each patient, we compared the worst histology result among colposcopically positive sites to the worst histology result among all sites biopsied, thereby more accurately determining the number of patients that would have been underdiagnosed by colposcopy than previously estimated. Materials and Methods: We utilized data from a clinical trial in which 850 diagnostic patients had been enrolled. Seven hundred and ninety-eight of the 850 patients had been examined by colposcopy, and biopsy samples were taken at colposcopically normal and abnormal sites. Our endpoints of interest were the percentages of patients underdiagnosed, and sensitivity and specificity of colposcopy. Results: With the threshold of low-grade squamous intraepithelial lesions for positive colposcopy and histology diagnoses, the sensitivity of colposcopy decreased from our previous assessment of 87.0% to 74.0%, while specificity remained the same. The drop in sensitivity was the result of histologically positive sites that were diagnosed as negative by colposcopy. Thus, 28.4% of the 798 patients in this diagnostic group would have had their condition underdiagnosed by colposcopy in the clinic. Conclusions: In utilizing biopsies at multiple sites of the cervix, we present a more precise methodology for determining the accuracy of colposcopy. The true accuracy of colposcopy is lower than previously estimated. Nevertheless, our results reinforce previous conclusions that colposcopy has an important role in the diagnosis of cervical precancer.
An Approach for the Adaptive Solution of Optimization Problems Governed by Partial Diﬀerential Equations with Uncertain Coeﬃcients
(2012-09-05) Kouri, Drew; Heinkenschloss, Matthias; Sorensen, Danny C.; Riviere, Beatrice M.; Cox, Dennis D.
Using derivative based numerical optimization routines to solve optimization problems governed by partial differential equations (PDEs) with uncertain coefficients is computationally expensive due to the large number of PDE solves required at each iteration. In this thesis, I present an adaptive stochastic collocation framework for the discretization and numerical solution of these PDE constrained optimization problems. This adaptive approach is based on dimension adaptive sparse grid interpolation and employs trust regions to manage the adapted stochastic collocation models. Furthermore, I prove the convergence of sparse grid collocation methods applied to these optimization problems as well as the global convergence of the retrospective trust region algorithm under weakened assumptions on gradient inexactness. In fact, if one can bound the error between actual and modeled gradients using reliable and efficient a posteriori error estimators, then the global convergence of the proposed algorithm follows. Moreover, I describe a high performance implementation of my adaptive collocation and trust region framework using the C++ programming language with the Message Passing interface (MPI). Many PDE solves are required to accurately quantify the uncertainty in such optimization problems, therefore it is essential to appropriately choose inexpensive approximate models and large-scale nonlinear programming techniques throughout the optimization routine. Numerical results for the adaptive solution of these optimization problems are presented.
Analyzing Single-Molecule Manipulation Experiments
(2008-10) Calderon, Christopher P.; Harris, Nolan C.; Kiang, Ching-Hwa; Cox, Dennis D.
Single-molecule manipulation studies can provide quantitative information about the physical properties of complex biological molecules without ensemble artifacts obscuring the measurements. We demonstrate computational techniques which aim at more fully utilizing the wealth of information contained in noisy experimental time series. The "noise" comes from multiple sources, e.g. inherent thermal motion, instrument measurement error, etc. The primary focus of this article is a methodology for using time domain based methods for extracting the effective molecular friction from single-molecule pulling data. We studied molecules composed of 8 tandem repeat titin I27 domains, but the modeling approaches have applicability to other single-molecule mechanical studies. The merits and challenges associated with applying such a computational approach to existing single-molecule manipulation data are also discussed.
Application of Bayesian Modeling in High-throughput Genomic Data and Clinical Trial Design
(2013-08-23) Xu, Yanxun; Cox, Dennis D.; Ji, Yuan; Qiu, Peng; Scott, David W.; Nakhleh, Luay K.
My dissertation mainly focuses on developing Bayesian models for high-throughput data and clinical trial design. Next-generation sequencing (NGS) technology generates millions of short reads, which provide valuable information for various aspects of cellular activities and biological functions. So far, NGS techniques have been applied in quantitatively measurement of diverse platforms, such as RNA expression, DNA copy number variation (CNV) and DNA methylation. Although NGS is powerful and largely expedite biomedical research in various fields, challenge still remains due to the high modality of disparate high-throughput data, high variability of data acquisition, high dimensionality of biomedical data, and high complexity of genomics and proteomics, e.g., how to extract useful information for the enormous data produced by NGS or how to effectively integrate the information from different platforms. Bayesian has the potential to fill in these gaps. In my dissertation, I will propose Bayesian-based approaches to address above challenges so that we can take full advantage of the NGS technology. It includes three specific topics: (1) proposing BM-Map: a Bayesian mapping of multireads for NGS data, (2) proposing a Bayesian graphical model for integrative analysis of TCGA data, and (3) proposing a non- parametric Bayesian Bi-clustering for next generation sequencing count data. For the clinical trial design, I will propose a latent Gaussian process model with application to monitoring clinical trials.
Applications of Bayesian sequential decision theory to medical decision-making
(2003) Swartz, Richard J.; Cox, Dennis D.
This thesis considers the use of Bayesian sequential decision theory for the diagnosis of pre-cancerous lesions of the cervix otherwise known as cervical intraepithelial neoplasia (CIN). We consider a sequence of n diagnostic tests where the ordering of the tests is predetermined. After each test in the sequence, the clinician must either make a treatment decision based on available information or continue testing. Our method allows the use of the previously collected information along with the new information collected at each level. In addition, we apply Bayesian sequential decision theory in a setting where the observations are not independent and identically distributed. Before this theory can be applied to the medical setting, a satisfactory method of attaining the costs of diagnostic tests and losses associated with treatment decisions must be specified. These costs and losses must be in the same units of measurement and they should include monetary considerations and both positive and negative patient outcomes. This thesis provides a method to determine bounds on relative costs and losses for medical decisions. First the medical decision process is modelled as a Bayesian sequential decision problem. Then we assume the current standard of care for detection of CIN is optimal, and use the model to determine bounds for the costs associated with testing and the losses associated with treatment. Unlike several other approaches, the costs and losses from our analysis potentially incorporate both monetary considerations and patient outcomes associated with testing and treatment or non-treatment. We estimate the probabilities necessary for the model from data collected at the University of Texas M. D. Anderson Cancer Center. We use both maximum likelihood estimates and Bayesian posterior mean estimates, with a prior developed from the literature. We also randomly sampled from the posterior distribution and compared our empirical bounds on the losses to values for the bounds on the losses reported in the cancer literature. The implications are discussed further in the thesis.
Aspects of functional data inference and its applications
(2006) Lee, Jong Soo; Cox, Dennis D.
We consider selected topics in estimation and testing of functional data. In many applications of functional data analysis, we aim to compare the sample functional data from two or more populations. However, the raw functional data often contains noise, some of which can be huge outliers. Hence, we must first perform the smoothing and estimation of functional data, but the existing methods for robust smoothing parameter selection are unsatisfactory. We present an efficient way to compute a smoothing parameter which can be generally applied to most robust smoothers. Then, we propose a procedure for testing pointwise difference of functional data in the two-sample framework. Our proposed method is a generalization of Hotelling's T2 test, and we utilize an adaptive truncation technique of Fan and Lin (1998) for dimension reduction and development of the test statistic. We show that our method performs well when compared with the existing testing procedures. Furthermore, we propose a method to detect the significantly different regions between curves. Once we determine that the samples curves from the two or more populations are significantly different overall, we want to look at the local regions of the curves and see where the differences occur. We present a modification of the multiple testing procedure of Westfall and Young (1993) for this testing method. Finally, we apply our proposed methods to the data from the fluorescence spectroscopic device. The fluorescence spectroscopic device is a medical device designed for early detection of cervical cancer, and the output from the device is a functional data, which makes the analysis challenging. The problems posed by this application have motivated the development of the methodologies in the present work, and we demonstrate that our methods work well in this application.
Bayesian decision-theoretic method and semi-parametric approach with applications in clinical trial designs and longitudinal studies
(2013-11-25) Jiang, Fei; Lee, J. Jack; Cox, Dennis D.; Scott, David W.; Ma, Yanyuan; Tapia, Richard A.
The gold of biostatistical researches is to develop statistical tools that improves human health or increases understanding of human biology. One area of the studies focuses on designing clinical trials to find out if new drugs or treatments are efficacious. The other area focuses on studying diseases related variables, which gives better understanding of the diseases. The thesis explores these areas from both theoretical and practical points of views. In addition, the thesis develop statistical devices which improve the existing methods in these areas. Firstly, the thesis proposes a Bayesian decision-theoretic group sequential – adaptive randomization phase II clinical trial design. The design improves the trial efficiency by increasing statistical power and reducing required sample sizes. The design also increases patients’ individual benefit, because it enhances patients’ opportunities of receiving better treatments. Secondly, the thesis develops a semiparametric restricted moment model and a score imputation estimation for survival analysis. The method is more robust than the parametric alternatives. In addition to data analysis, the method is used to design a seamless phase II/III clinical trial. The seamless phase II/III clinical trial design shortens the durations between phase II and III studies, and improves the efficiency of the traditional designs by utilizing additional short term information for making decisions. Finally, the thesis develops a partial linear time varying semi-parametric single-index risk score model and a fused B-spline/kernel estimation for longitudinal data analysis. The method models confounder effects linearly. In addition, it uses a nonparametric nonlinear function, namely the single-index risk score, to model the effects of interests. The fused B-spline/kernel technique estimates both the parametric and nonparametric components consistently. The methodology is applied to study the onsite of Huntington’s disease in determining certain time varying covariate effects on the disease risk.
Bayesian semiparametric and flexible models for analyzing biomedical data
(2010) Leon Novelo, Luis G.; Cox, Dennis D.
In this thesis I develop novel Bayesian inference approaches for some typical data analysis problems as they arise with biomedical data. The common theme is the use of flexible and semi-parametric Bayesian models and computation intensive simulation-based implementations. In chapter 2, I propose a new approach for inference with multivariate ordinal data. The application concerns the assessment of toxicities in a phase III clinical trial. The method generalizes the ordinal probit model. It is based on flexible mixture models. In chapter 3, I develop a semi-parametric Bayesian approach for bio-panning phage display experiments. The nature of the model is a mixed effects model for repeated count measurements of peptides. I develop a non-parametric Bayesian random effects distribution and show how it can be used for the desired inference about organ-specific binding. In chapter 4, I introduce a variation of the product partition model with a non-exchangeable prior structure. The model is applied to estimate the success rates in a phase II clinical of patients with sarcoma. Each patient presents one subtype of the disease and subtypes are grouped by good, intermediate and poor prognosis. The prior model respects the varying prognosis across disease subtypes. Two subtypes with equal prognoses are more likely a priori to have similar success rates than two subtypes with different prognoses.
Essays in Structural Econometrics of Auctions
(2012-09-05) Bulbul Toklu, Seda; Sickles, Robin C.; Medlock, Kenneth B., III; Cox, Dennis D.
The first chapter of this thesis gives a detailed picture of commonly used structural estimation techniques for several types of auction models. Next chapters consist of essays in which these techniques are utilized for empirical analysis of auction environments. In the second chapter we discuss the identification and estimation of the distribution of private signals in a common value auction model with an asymmetric information environment. We argue that the private information of the informed bidders are identifiable due to the asymmetric information structure. Then, we propose a two stage estimation method, which follows the identification strategy. We show, with Monte-Carlo experiments, that the estimator performs well. Third chapter studies Outer Continental Shelf drainage auctions, where oil and gas extraction leases are sold. Informational asymmetry across bidders and collusive behavior of informed firms make this environment very unique. We apply the technique proposed in the second chapter to data from the OCS drainage auctions. We estimate the parameters of a structural model and then run counterfactual simulations to see the effects of the informational asymmetry on the government's auction revenue. We find that the probability that information symmetry brings higher revenue to the government increases with the value of the auctioned tract. In the fourth chapter, we make use of the results in the multi-unit auction literature to study the Balancing Energy Services auctions (electricity spot market auctions) in Texas. We estimate the marginal costs of bidders implied by the Bayesian-Nash equilibrium of the multi-unit auction model of the market. We then compare the estimates to the actual marginal cost data. We find that, for the BES auction we study, the three largest bidders, Luminant, NRG and Calpine, have marked-down their bids more than the optimal amount implied by the model for the quantities where they were short of their contractual obligations, while they have put a mark-up larger than the optimal level implied by the model for quantities in excess of their contract obligations. Among the three bidders we studied, Calpine has come closest to bidding its optimal implied by the Bayesian-Nash equilibrium of the multi-unit auction model of the BES market.
Estimating nonlinear functionals of a random field
(2001) Boeckenhauer, Rachel Kaye; Cox, Dennis D.
Environmental data are gathered with the goal of estimating some quantity of interest. In particular, in the case of groundwater or soil contamination, it is desirable to estimate the total amount of contaminant present within a region in order to more effectively remediate the contamination. This problem has been generally unaddressed previously, and is of interest to environmental scientists. A method is introduced here to estimate the integral of a lognormal process over a region using Monte Carlo simulation of the process conditional on the observed data. The performance of the method is evaluated with an application to groundwater data. Results are compared using uniform sampling and importance sampling over the region.
Functional Data Analysis on Spectroscopic Data
(2016-01-28) Wang, Lu; Cox, Dennis D.; Scott, David W; Zhang, Yin
Cervical cancer is a very common type of cancer that is highly curable if treated early. We are investigating spectroscopic devices that make in-vivo cervical tissue measurements to detect pre-cancerous and cancerous lesions. This dissertation is focused on new methods and algorithms to improve the performance of the device, treating the spectroscopic measurements as functional data. The first project is to calibrate the device measurements using correction factors from a log additive model, based on results from a carefully designed experiment. The second project is a peak finding algorithm using local polynomial regression to get accurate peak location and height estimates of one of the standards (Rhodamine B) measurements from the experiment. We propose a plug-in bandwidth selection method to estimate curve peak location and height. Simulation results and asymptotic properties are presented. The third project is based on patient measurements, particularly when the diseased and non-diseased cases are highly unbalanced. A marginalized corruption methodology is introduced to improve the classification results. Performance of several classification methods is compared.
Functional data classification and covariance estimation
(2009) Zhu, Hongxiao; Cox, Dennis D.
Focusing on the analysis of functional data, the first part of this dissertation proposes three statistical models for functional data classification and applies them to a real problem of cervical pre-cancer diagnosis; the second part of the dissertation discusses covariance estimation of functional data. The functional data classification problem is motivated by the analysis of fluorescence spectroscopy, a type of clinical data used to quantitatively detect early-stage cervical cancer. Three statistical models are proposed for different purposes of the data analysis. The first one is a Bayesian probit model with variable selection, which extracts features from the fluorescence spectroscopy and selects a subset from these features for more accurate classification. The second model, designed for the practical purpose of building a more cost-effective device, is a functional generalized linear model with selection of functional predictors. This model selects a subset from the multiple functional predictors through a logistic regression with a grouped Lasso penalty. The first two models are appropriate for functional data that are not contaminated by random effects. However, in our real data, random effects caused by devices artifacts are too significant to be ignored. We therefore introduce the third model, the Bayesian hierarchical model with functional predictor selection, which extends the first two models for this more complex data. Besides retaining high classification accuracy, this model is able to select effective functional predictors while adjusting for the random effects. The second problem focused on by this dissertation is the covariance estimation of functional data. We discuss the properties of the covariance operator associated with Gaussian measure defined on a separable Hilbert Space and propose a suitable prior for Bayesian estimation. The limit of Inverse Wishart distribution as the dimension approaches infinity is also discussed. This research provides a new perspective for covariance estimation in functional data analysis.
Inverse decision theory with medical applications
(2005) Davies, Kalatu R.; Cox, Dennis D.
Medical decision makers would like to use decision theory to determine optimal treatment strategies for patients, but it can be very difficult to specify loss functions in the medical setting, especially when trying to assign monetary value to health outcomes. These issues led to the development of an alternative approach, called Inverse Decision Theory (IDT), in which given a probability model and a specific decision rule, we determine the set of losses for which that decision rule is optimal. This thesis presents the evolution of the IDT method and its applications to medical treatment decision rules. There are two ways in which we can use the IDT method. Under the first approach, we operate under the assumption that the decision rule of interest is optimal, and use the prior information that we have to make inferences on the losses. The second approach involves the use of the prior information to derive an optimal region and determine if the losses in this region are reasonable based on our prior information. We illustrate the use of IDT by applying it to the current standard of care (SOC) for the detection and treatment of cervical neoplasias. First, we model the diagnostic and treatment process as a Bayesian sequential decision procedure. Then, we determine the Bayes risk expression for all decision rules and compare the Bayes risk expression for the current SOC decision rule to the Bayes risk expressions of all other decision rules, forming linear inequality constraints on a region under which the current SOC is optimal. The current standard of care has been in use for many years, but we find another decision rule to be optimal. We question whether the current standard of care is the optimal decision rule and will continue to examine these implications and the practicality of implementing this new decision rule. The IDT method provides us with a mathematical technique for dealing with the challenges in formally quantifying patient experiences and outcomes. We believe that it will be applicable to many other disease conditions and become a valuable tool for determining optimal medical treatment standards of care.
Investigation of the tau-leap method for stochastic simulation
(2008) Noyola-Martinez, Josue C.; Cox, Dennis D.
The use of the relatively new tau-leap algorithm to model the kinematics of regulatory systems and other chemical processes inside the cell, is of great interest; however, the accuracy of the tau-leap algorithm is not known. We introduce a new method that enables us to establish the accuracy of the tau-leap method effectively. Our approach takes advantage of the fact that the stochastic simulation algorithm (SSA) and the tau-leap method can be represented as a special type of counting process, that can essentially "couple", or tie together, a single realization of the SSA process to one of the tau-leap. Because the SSA is exact we can evaluate the accuracy of the tau-leap by comparing it to the SSA. Our approach gives error estimates which are unrivaled by any method currently available. Moreover, our coupling algorithm allows us to propose an adaptive parameter selection algorithm which finds the appropriate parameter values needed to achieve a pre-determined error threshold in the tau-leap algorithm. This error-controlled adaptive parameter selection method could not have been proposed before the introduction of our coupling algorithm, and it is a novel approach to the use of the tau-leap algorithm.
Limiting Approximations for Stochastic Processes in Systems Biology
(2015-04-24) Woroszylo, Casper; Cox, Dennis D.; Kimmel, Marek; Levine, Herbert
Interest in stochastic modeling of biochemical processes has increased over the past two decades due to advancements in computing power and an increased understanding of the underlying physical phenomena. The Gillespie algorithm is an exact simulation technique for reproducing sample paths from a continuous-time Markov chain. However, when spatial and temporal time scales vary within a given system, a purely stochastic approach becomes intractable. In this work, we develop two types of hybrid approximations, namely piecewise-deterministic approximations. These approaches yield strong approximations for either the entire biochemical system or a subset of the system, provided the purely stochastic system is appropriately rescaled.
Multi-scale behavior in chemical reaction systems: Modeling, applications, and results
(2008) Turner, Jesse Hosea, III; Cox, Dennis D.
Four major approaches model the time dependent behavior of chemical reaction systems: ordinary differential equations (ODE's), the &tgr;-leap algorithm, stochastic differential equations (SDE's), and Gillespie's stochastic simulation algorithm (SSA). ODE's are simulated the most quickly of these, but are often inaccurate for systems with slow rates and molecular species present in small numbers. Under ideal conditions, the SSA is exact, but computationally inefficient. Unfortunately, many reaction systems exhibit characteristics not well captured individually by any of these methods. Therefore, hybrid models incorporating aspects from all four must be employed. The aim is to construct an approach that is close in accuracy to the SSA, useful for a wide range of reaction system examples, and computationally efficient. The Adaptive Multi-scale Simulation Algorithm (AMSA) uses the SSA for slow reactions, SDE's for medium-speed reactions, ODE's for fast reactions, and the tau-leap algorithm for non-slow reactions involving species small in number. This article introduces AMSA and applies it to examples of reaction systems involving genetic regulation. A thorough review of existing reaction simulation algorithms is included. The computational performance and accuracy of AMSA's molecular distributions are compared to those of the SSA, which is used as the golden standard of accuracy. The use of supercomputers can generate much larger data sets than serial processors in roughly the same amount of computational time. Therefore, multi-processor machines are also employed to assess the accuracy of AMSA simulations.
Multilevel classification: Classification of populations from measurements on members
(2007) Yamal, Jose-Miguel; Cox, Dennis D.
Multilevel classification is a problem in statistics which has gained increasing importance in many real-world problems, but it has not yet received the same statistical understanding as the general problem of classification. An example we consider here is to develop a method to detect cervical neoplasia (pre-cancer) using quantitative cytology, which involves measurements on the cells obtained in a Papanicolou smear. The multilevel structure comes from the embedded cells within a patient, where we have quantitative measurements on the cells, yet we want to classify the patients, not the cells. An additional challenge comes from the fact that we have a high-dimensional feature vector of measurements on each cell. The problem has historically been approached in two ways: (a) ignore this multilevel structure of the data and perform classification at the microscopic (cellular) level, and then use ad-hoc methods to classify at the macroscopic (patient) level, or (b) summarize the microscopic level data using a few statistics and then use these to compare the subjects at the macroscopic level. We consider a more rigorous statistical approach, the Cumulative Log-Odds (CLO) Method, which models the posterior log-odds of disease for a patient given the cell-level measured feature vectors for that patient. Combining the CLO method with a latent variable model (Latent-Class CLO Method) helps to account for between-patient heterogeneity. We apply many different approaches and evaluate their performance using out of sample prediction. We find that our best methods classify with substantial greater accuracy than the subjective Papanicolou Smear interpretation by a clinical pathologist.
Nonparametric density contour estimation
(1998) Gebert, Mark Allen; Cox, Dennis D.
Estimation of the level sets for an unknown probability density is done with no specific assumed form for that density, that is, non-parametrically. Methods for tackling this problem are presented. Earlier research showed existence and properties of an estimate based on a kernel density estimate in one dimension. Monte Carlo methods further demonstrated the reasonability of extending this approach to two dimensions. An alternative procedure is now considered that focuses on properties of the contour itself; procedures wherein we define and make use of an objective function based on the characterization of contours as enclosing regions of minimum area given a constraint on probability. Restricting our attention to (possibly non-convex) polygons as candidate contours, numeric optimization of this difficult non-smooth objective function is accomplished using pdsopt, for Parallel Direct Search OPTimization, a set of routines developed for minimization of a scalar-valued function over a high-dimensional domain. Motivation for this method is given, as well as results of simulations done to test it; these include exploration of a Lagrange-multiplier penalty on area and the need which arises for addition of a penalty on the "roughness" of a polygonal contour.

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